In this paper, we prove a criterion for the irrationality of certain constants which arise from the Ramanujan summation of a family of infinite divergent sums. As an application, we provide a sufficient criterion for the irrationality of the values of the Riemann zeta function in the interval (0, 1). We further see that our discussion leads to a natural generalization of a result of Sondow on the irrationality criterion for the Euler-Mascheroni constant.